The backlog and depletion-time process for M/G/1 vacation models with exhaustive service discipline

The vacation model studied is an M/G/1 queueing system in which the server attends iteratively to ‘secondary' or ‘vacation' tasks at ‘primary' service completion epochs, when the primary queue is exhausted. The time-dependent and steady-state distributions of the backlog process [6] are obtained via their Laplace transforms. With this as a stepping stone, the ergodic distribution of the depletion time [5] is obtained. Two decomposition theorems that are somewhat different in character from those available in the literature [2] are demonstrated. State space methods and simple renewal-theoretic tools are employed.

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M/G/1 vacation model with limited service discipline and hybrid switching-on policy

Mathematical Problems in Engineering ◽

10.1155/s1024123x97000550 ◽

1997 ◽

Vol 3 (3) ◽

pp. 243-253

Author(s):

Alexander V. Babitsky

Keyword(s):

Generating Function ◽

Busy Period ◽

Optimization Problem ◽

Queueing System ◽

Service Discipline ◽

Multiple Vacations ◽

Queueing Process ◽

Vacation Model ◽

Hybrid Switching ◽

Limited Service

The author studies an M/G/1 queueing system with multiple vacations. The server is turned off in accordance with the K-limited discipline, and is turned on in accordance with the T-N-hybrid policy. This is to say that the server will begin a vacation from the system if either the queue is empty orKcustomers were served during a busy period. The server idles until it finds at leastNwaiting units upon return from a vacation.Formulas for the distribution generating function and some characteristics of the queueing process are derived. An optimization problem is discussed.

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Time-dependent process of M/G/1 vacation models with exhaustive service

Journal of Applied Probability ◽

10.1017/s0021900200043163 ◽

1992 ◽

Vol 29 (02) ◽

pp. 418-429 ◽

Cited By ~ 3

Author(s):

Hideaki Takagi

Keyword(s):

Steady State ◽

Initial Conditions ◽

Setup Time ◽

Time Dependent ◽

Time Model ◽

Vacation Models ◽

Single Vacation ◽

Multiple Vacation ◽

Virtual Waiting Time ◽

Vacation Model

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

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Time-dependent process of M/G/1 vacation models with exhaustive service

Journal of Applied Probability ◽

10.2307/3214578 ◽

1992 ◽

Vol 29 (2) ◽

pp. 418-429 ◽

Cited By ~ 16

Author(s):

Hideaki Takagi

Keyword(s):

Steady State ◽

Initial Conditions ◽

Setup Time ◽

Time Dependent ◽

Time Model ◽

Vacation Models ◽

Single Vacation ◽

Multiple Vacation ◽

Virtual Waiting Time ◽

Vacation Model

Generalized M/G/1 vacation systems with exhaustive service include multiple and single vacation models and a setup time model possibly combined with an N-policy. In these models with given initial conditions, the time-dependent joint distribution of the server's state, the queue size, and the remaining vacation or service time is known (Takagi (1990)). In this paper, capitalizing on the above results, we obtain the Laplace transforms (with respect to time) for the distributions of the virtual waiting time, the unfinished work (backlog), and the depletion time. The steady-state limits of those transforms are also derived. An erroneous expression for the steady-state distribution of the depletion time in a multiple vacation model given by Keilson and Ramaswamy (1988) is corrected.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.2307/3214324 ◽

1989 ◽

Vol 26 (1) ◽

pp. 142-151 ◽

Cited By ~ 2

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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On a non-Markovian time-dependent queueing system

Journal of Applied Probability ◽

10.1017/s0021900200041875 ◽

1989 ◽

Vol 26 (01) ◽

pp. 142-151

Author(s):

S. D. Sharma

Keyword(s):

Steady State ◽

Generating Function ◽

Queue Length ◽

Queueing System ◽

Length Distribution ◽

Probability Generating Function ◽

Transient State ◽

Time Dependent ◽

Queue Length Distribution ◽

State Behaviour

This paper studies the transient and steady-state behaviour of a continuous and discrete-time queueing system with non-Markovian type of departure mechanism. The Laplace transforms of the probability generating function of the time-dependent queue length distribution in the transient state are obtained and the probability generating function of the queue length distribution in the steady state is derived therefrom. Finally, some particular cases are discussed.

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L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations

Journal of Scientific Computing ◽

10.1007/s10915-021-01416-z ◽

2021 ◽

Vol 87 (1) ◽

Author(s):

Yanlai Chen ◽

Lijie Ji ◽

Akil Narayan ◽

Zhenli Xu

Keyword(s):

Steady State ◽

Nonlinear Equations ◽

Time Dependent

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Ground- and excited-state characteristics in photovoltaic polymer N2200

RSC Advances ◽

10.1039/d1ra01474a ◽

2021 ◽

Author(s):

Guanzhao Wen ◽

Xianshao Zou ◽

Rong Hu ◽

Jun Peng ◽

Zhifeng Chen ◽

...

Keyword(s):

Density Functional Theory ◽

Excited State ◽

Steady State ◽

Excited States ◽

Density Functional ◽

Density Functional Theory Calculations ◽

Time Dependent ◽

Functional Theory ◽

Time Resolved ◽

Dependent Density

Ground- and excited-states properties of N2200 have been studied by steady-state and time-resolved spectroscopies as well as time-dependent density functional theory calculations.

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Explicit steady state solutions for a particularM(x)/M/1 queueing system

Naval Research Logistics Quarterly ◽

10.1002/nav.3800240412 ◽

1977 ◽

Vol 24 (4) ◽

pp. 651-659 ◽

Cited By ~ 4

Author(s):

George L. Jensen ◽

Albert S. Paulson ◽

Pasquale Sullo

Keyword(s):

Steady State ◽

Queueing System ◽

Steady State Solutions

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Determinants of mRNA stability in Dictyostelium discoideum amoebae: differences in poly(A) tail length, ribosome loading, and mRNA size cannot account for the heterogeneity of mRNA decay rates.

Molecular and Cellular Biology ◽

10.1128/mcb.8.5.1957 ◽

1988 ◽

Vol 8 (5) ◽

pp. 1957-1969 ◽

Cited By ~ 23

Author(s):

R A Shapiro ◽

D Herrick ◽

R E Manrow ◽

D Blinder ◽

A Jacobson

Keyword(s):

Steady State ◽

Dictyostelium Discoideum ◽

Mrna Decay ◽

Decay Rates ◽

Time Dependent ◽

Translational Efficiency ◽

Cdna Clones ◽

Dependent Manner ◽

Significant Difference ◽

Kinetics Of

As an approach to understanding the structures and mechanisms which determine mRNA decay rates, we have cloned and begun to characterize cDNAs which encode mRNAs representative of the stability extremes in the poly(A)+ RNA population of Dictyostelium discoideum amoebae. The cDNA clones were identified in a screening procedure which was based on the occurrence of poly(A) shortening during mRNA aging. mRNA half-lives were determined by hybridization of poly(A)+ RNA, isolated from cells labeled in a 32PO4 pulse-chase, to dots of excess cloned DNA. Individual mRNAs decayed with unique first-order decay rates ranging from 0.9 to 9.6 h, indicating that the complex decay kinetics of total poly(A)+ RNA in D. discoideum amoebae reflect the sum of the decay rates of individual mRNAs. Using specific probes derived from these cDNA clones, we have compared the sizes, extents of ribosome loading, and poly(A) tail lengths of stable, moderately stable, and unstable mRNAs. We found (i) no correlation between mRNA size and decay rate; (ii) no significant difference in the number of ribosomes per unit length of stable versus unstable mRNAs, and (iii) a general inverse relationship between mRNA decay rates and poly(A) tail lengths. Collectively, these observations indicate that mRNA decay in D. discoideum amoebae cannot be explained in terms of random nucleolytic events. The possibility that specific 3'-structural determinants can confer mRNA instability is suggested by a comparison of the labeling and turnover kinetics of different actin mRNAs. A correlation was observed between the steady-state percentage of a given mRNA found in polysomes and its degree of instability; i.e., unstable mRNAs were more efficiently recruited into polysomes than stable mRNAs. Since stable mRNAs are, on average, "older" than unstable mRNAs, this correlation may reflect a translational role for mRNA modifications that change in a time-dependent manner. Our previous studies have demonstrated both a time-dependent shortening and a possible translational role for the 3' poly(A) tracts of mRNA. We suggest, therefore, that the observed differences in the translational efficiency of stable and unstable mRNAs may, in part, be attributable to differences in steady-state poly(A) tail lengths.

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